5 REPRESENTATIONS OF ALGEBRAS WITH INVOLUTION 347 A representation SH* U(s) of A in H is said to be topologically irreducible if there exists no closed vector subspace E of H, other than {0} and H, which is stable with respect to U. From (15.5.3) we obtain the following irreducibility criterion: (15.5.4) In order that U should be topologically irreducible, it is necessary and sufficient that the only orthogonal projections P such that PU(s) = U(s)P for all s e A should be 0 and 1H . For this condition expresses that {0} and H are the only closed subspaces of H which are stable under U. (15.5.5) Let SH* U(s) be a representation of A in H, let E be the closure in H of the vector subspace generated by the elements U(s) • x for s e A and x e H, and let E' be the set of all x ell such that U(s) - x = 0 for all s 6 A. Then E and E' are stable with respect to U and are orthogonal supplements of each other in H. Since U(sf) = U(s)U(t), it is clear that E and E' are stable subspaces of H. Let E" be the orthogonal supplement of E in H. We have seen earlier that E" is stable with respect to U. But if x e E", we have 17(5-) • x e E by definition, hence U(s) • x e E n E" = {0} for all s e A, so that E" <= E'. Conversely, if x e E', s e A and y e H, we have (x \ U(s) • y) = (U(s*) • x \ y) = 0 by defini- tion, so that x is orthogonal to E, and therefore E' cz E". The subspace E is called the essential subspace for U. If E' = {0}, the representation U is said to be nondegenerate. An equivalent condition is that the elements U(s) - x should form a total subspace of H; by (15.5.2), this is always the case if A has a unit element. A vector x0 e H is a totalizer or totalizing vector for a representation U of A in H if the vector subspace of H generated by the transforms U(s) • x0 of XQ , as s runs through A, is dense in H. (In any case, this subspace is stable with respect to U.) A representation which admits a totalizer is said to be topologi- cally cyclic. If U is topologically irreducible, every nonzero vector x0 e H is a totalizer, and conversely. (15.5.6) Suppose that A has a unit element. Let s\-+ U(s) be a nondegenerate representation of A in a separable Hilbert space H. Then H is the Hilbert sum of a sequence (Hn)( finite or infinite) of closed subspaces,, stable with respect to U and such that the restriction of U to each Hn is topologically cyclic.(s) • x = £ Un(s) - xn, and